Dipping a Toe in the Collatz Intoxicant | Steve Wait – April 30, 2022

Having resisted temptation, I have succumbed to the sirens’ song of “Come on in. The water is fine.”. The following reflects my in… and out for now approach.

In the following:

If n is odd, then the result of a 3n+1 is even and the formula is rewritten as $\frac{{3}{n}+{1}}{{2}}$ 

If n is even, then $\frac{{n}}{{2}}$ is rewritten as $\frac{{1}{n}+{0}}{{2}}$

Thus, in my framework $\frac{{3}{n}+{1}}{{2}}$ and $\frac{{1}{n}+{0}}{{2}}$ are representative of the Collatz Conjecture.

Note: In the following table, sequences are extended and linear simplification is performed via y=mx+b

Note: The angle of the lines being $θ={{tan}}^{-{1}}{{m}}$ establishes their convergence at x-0.50, y-.25

For the integer solution lines shown in Fig. 1, y outputs hailstone between y=0.50x and y=1.50x+0.50 as well for the transpose of y=-0.50x-0.50 and y=1.50x-1.00

Negative x integers seem either to loop for the former or terminate for the latter. Extending the line sequence yields convergence along the axis of symmetry; θ of odd/even pairings decreases and y outputs race toward infinity.

Only the input of an even x integer on the even line is valid for a y integer output. Only the input of an odd x integer on the odd line is valid for a y integer output.

Integer solution table as follow:

A color coded graphic of the above table: